coherent oscillator injection and recovery

Coherent oscillators—astronomical sources that pulse or oscillate in a phase-stable way over long timescales—have been useful astrophysical tools. For two examples: Stably pulsing pulsars were used to discover gravitational radiation (and are being used to find the stochastic background). Delta-scuti star asteroseismic modes were used to find orbital companions. This led undergrad Nana Miller (NYU) and me and others to look for all the coherent modes we can find among all the stars in the NASA Kepler Mission data. We have have technology to find modes, and we have technology to test for coherence. Now we have to do injection–recovery tests to estimate our detection limits. Today I delivered a simple plan for doing injections.

Our plan is to produce a catalog, not of stars but of modes, every one of which has a coherence time that is longer than the lifetime (4 years) of the Kepler Mission. Then: What do we use them for?

double periodogram

Cole Johnston (Leuven) is in New York this week. We discussed the problem of finding oscillation modes in the photometry of stars in the presence of a large, binary-induced periodicity. What he kind-of wants is a simultaneous fitting of a flexible periodic function plus a periodogram. We did some experiments (very promising!) and discussed the elements that will come together to make this all happen. The final method will look like a double fourier transform, in which one frequency grid gets the periodic part, and the other grid gets the rest of the modes and noise.

an alternative to the L–S periodogram

Following some experiments and rants over the last few days with Nora Eisner (Flatiron), I wrote down today an algorithm for a hacky replacement of the Lomb–Scargle periodogram. This periodogram method has various bad pathologies, the worst of which is that it presumes that there is exactly one frequency that fully generates the data. If there are two, the assumptions are broken and the good properties are lost.

Not that my alternative has any good properties! It is like the radio interferometry method called CLEAN: It involves iteratively identifying frequencies and fitting them out. It's terrible. But it might be better than some of the wacky hacks that people do right now in the asteroseismology community.

Phi-M radio

I worked today with Abby Shaum (CUNY) on her paper about her phase-demodulator to find exoplanet and substellar companions to stars by the timing of asteroseismic modes. I suggested that we highlight the incredible simplicity of her project by writing the method as an algorithm of just a few lines.

phase and frequency variations

If a star has a (relatively) coherent oscillation mode, and you can monitor it over a long period of time, then orbital motion of the star can be seen as either phase or frequency variations of that mode. I've been working on this in different collaborations, with Dan Hey, with Simon J Murphy, with Abby Shaum (CUNY), and recently with Nora Eisner (Flatiron). Right now, Shaum, Eisner, and I are looking at signal-processing approaches that look like demodulators. What I'm interested in—at least in terms of me learning about signal processing—is how can we make a demodulator that demodulates both phase and frequency simultaneously. There must be hybrid and combined approaches. I'm also interested in what we can measure from incoherent oscillators.

simulating data for phase and frequency modulation

Abby Shaum (CUNY) and I have been working on phase demodulation for binary detection and characterization, using coherent oscillation modes in stellar light curves. We are taking a pure signal-processing approach, which is lightweight and fast, such that we could automatically apply it to everything in Kepler or TESS. We also want to do frequency modulation for incoherent modes (which somehow our people think won't work; won't it?).

Today we discussed how to build fake data to fully test our systems. In the coherent case, this is easy! In the incoherent case this is harder. We discussed simulating it by drawing from a Gaussian Process. And we discussed simulating it by forward modeling a stochastically forced, damped harmonic oscillator.

time dependences in ultra-short-period planets

I had a great conversation with Noah Sodickson (high schooler) about ultra-short planets in the NASA Kepler data. He has been filtering the light curves, running astronomers'-favorite “box least squares” and plotting the folded light curves. When he does this in quarters (meaning, cutting the data into 90-day chunks), he can see small variations in the transit shapes. These have been attributed to various things. My goal is to figure out a way to model these variations without averaging the data. Binning is sinning, after all. Sodickson can see that everything we see is a strong function of how we filter the light curves, so we have to think pretty hard about that.

signal processing vs forward modeling

Abby Shaum (CUNY) and I are trying to write up a paper about our work treating oscillating stars as something like FM radios: We use the oscillation modes as carrier frequencies and find any orbital companions through phase or frequency variations of that carrier signal. Today we discussed the difference between doing that and forward modeling the signal. The former is signal processing. The latter is a generative model. Very different! And in many senses forward modeling is more principled. But I still think (and hope) that signal processing has a place in astronomy.

radio reboot

[Somehow this blog keeps failing. I will try to get back into it, but no promises! I apologize to my loyal reader.]

Today I met with Abby Shaum (IPAC) who worked with me a few years ago making a phase demodulator to find stellar companions. The idea is that if a star is broadcasting a coherent (or even incoherent) asteroseismic or pulsation mode, and if the star is orbiting a companion, the kinematics of the orbit will be imprinted on phase and frequency modulations of the carrier frequency. Like a radio! Indeed we built a signal-processing method that looks just like a radio demodulator. Today we discussed how to reboot this project and write a paper for the refereed literature.

band-limited image models

I had my quasi-weekly call with Dustin Lang (Perimeter) today, in which we discussed how to represent the differences between images, or represent models of images. I am obsessing about how to model distortions of spectrograph calibration images (like arcs and flats), and Lang and I endlessly discuss difference imaging to discover time-variable phenomena and moving objects. One thing Lang has observed is that sometimes a parametric model of a reference image makes a better reference image for image differencing than the reference image itself. Presumably this is because the model de-noises the data? But if so, could we just use a band-limited non-parametric model say? We discussed many things related to all that.

objective functions and Nyquist sampling

Adrian Price-Whelan and I discussed today some oddities that Matt Daunt (NYU) is finding while trying to measure radial velocities in extremely noisy, fast APOGEE sub-exposures. He finds that the objective function we are using is not obviously smooth on 10-ish km/s velocity scales. Why not? We don't know. But what we do know is that a spectrograph with resolution 22,500 cannot put sharp structures into a likelihood function on scales smaller than about 13 km/s.

There's a nice paradox here, in fact: The spectrograph can't see features on scales smaller than 13 km/s, and yet we can reliably measure radial velocities much better than this! How? The informal answer is that the radial-velocity precision is 13 km/s divided by a certain, particular signal-to-noise. The formal answer involves information theory—the Fisher information, to be precise.

low-pass filter for non-uniformly sampled data

Adrian Price-Whelan (Flatiron) and I used the new fiNUFFT non-uniformly-sampled fast Fourier tranform code to build a low-pass filter for stellar spectra today. The idea is: There can't be any spectral information in the data at spectral resolutions higher than the spectrograph resolution. So we can low-pass filter in the log-wavelength domain and that should enforce finite spectral resolution. The context is: Making features to use in a regression or other machine-learning method. I don't know, but I think this is a rare thing: A low-pass filter that doesn't require uniformly or equally-spaced sampling in the x direction or time domain.

Lomb–Scargle and The Joker are the same at zero eccentricity

As my loyal reader knows, our radial-velocity companion-search code called The Joker is a brute-force least-square fitting (plus some Bayes) of an elliptical orbit at every possible period, orientation, and phase. In astrophysics, the Lomb–Scargle periodogram is a workhorse tool that, under the hood, is a brute-force least-square fitting of a sinusoid at every possible period and phase. So these two ideas are fundamentally incredibly similar. And indeed, today Winston Harris (MTSU) demonstrated quantitatively that these are the same, and will become identical as eccentricities go to zero. That's interesting, because if we are doing companion search and we don't mind making the approximation that eccentricities are zero, the fitting of sinusoids (even at arbitrary phase, because: trig identities) is way, way faster than the fitting of Kepler functions.

how precisely can you locate K freqencies?

In Fourier transforms, or periodograms, or signal processing in general, when you look at a time stream that is generated by a single frequency f (plus noise, say) and has a total time length T, you expect the power-spectrum ppeak, or the likelihood function for f to have a width that is no narrower than 1/T in the frequency direction. This is for extremely deep reasons, that relate to—among other things—the uncertainty principle. You can't localize a signal in both frequency and time simultaneously.

Ana Bonaca (Harvard) and I are fitting combs of frequencies to light curves. That is, we are fitting a model with K frequencies, equally spaced. We are finding that the likelihood function has a peak that is a factor of K narrower than 1/T, in both the central-frequency direction, and the frequency-spacing direction. Is this interesting or surprising? I have ways to justify this point, heuristically. But is there a fundamental result here, and where is it in the literature?

information and asteroseismology

When you measure the asteroseismic spectrum of a star, there are many observables: Mode frequencies, mode amplitudes, a frequency difference between modes, an envelope in frequency space where the modes have large amplitudes, and some more detailed things like even-odd-ell mode splittings and rotational splittings. Some of these observables are used in predicting or inferring stellar propeties (like mass, age, and composition) and some are not. Why not? My feeling (both from looking at the data and from thinking about the theory) is that the amplitudes of the modes (or mean amplitude over a set of modes) and the width of the envelope at nu-max are both going to be informative about stellar structure. Ana Bonaca (Harvard) and I discussed these things today, and how we might both measure and exploit this information with new data.

exoplanet transit science

I have had a good day finishing up my reading of the PhD dissertation of Emily Sandford (Columbia), who has a great collection of results on transit measurements of planets and stars. She makes use of Bayesian methods and also classic optimization or signal-processing methods to make challenging inferences, and she shows the limits and capabilities of current and future data. One thing I liked was that she works on what are called “single transits” where only one transit is found in the whole survey duration; what can you infer from that? A lot! (I have also worked on this problem long ago.) In the dissertation she busts a myth that the multiplicity distribution requires that there be a mixture of two qualitatively different kinds of planetary systems. I enjoyed that, and it leads to a lot of other science. Plus some creative work on understanding the detailed properties of multi-planet systems, treating them like sequence data. It's a great thesis and I am very much looking forward to tomorrow's defense.

asteroseismic likelihood

Ana Bonaca (Harvard) and I discussed two important successes today (both of them due to Bonaca): We found the issue in our code that caused there to be frequency structure on frequency scales smaller than what you might call the “uncertainty-principle” limit of the inverse of the duration of the survey. It was literally a bug, not a think-o. Which is reassuring. And we do have likelihoods that seem to peak (at least locally) at sensible values of delta-nu, the large frequency difference, in the parlance of asteroseismology. So now there are questions of how to improve performance, and how Bayesian to be, as it were. Frequentism is easier in some respects, harder in others. Lots of engineering to think about!

funding! and interpolation!

Kate Storey-Fisher (NYU) got great news today. Her NASA FINESST proposal was funded! So she gets paid for the next three years. And she did it herself. Funding is one of the most challenging, confusing, disheartening, and complicated parts of this job. I don't like it, but it's reality. Congratulations, Storey-Fisher

On my student-research call, I worked on interpolating complex numbers with Avery Simon (NYU). We have a set of phase transforms of two very similar images, which return, for components, complex amplitudes (or amplitudes and phases, if you like). Now the question is: How to interpolate between these images, or their phase transforms? We want to test the hypothesis (put forward in computer-vision research like this) that interpolating in complex phase is way better in many respects than interpolating naively.

#AAS235 poster on demodulation

Today I helped Abby Shaum (NYU) finish her poster for the #AAS235 Meeting on using signal processing—and in particular phase demodulation—to find binary companions to stars with coherent asteroseismic modes in the NASA Kepler Mission data. Our abstract ended up being different from what we submitted! I hope no-one minds. But she has a beautiful result, and a pretty mechanistic method to search for binaries. Now our next thing is to figure out how to generalize the method to modes that aren't coherent over the full mission lifetime. If we can do that, there is a lot more we can find, I suspect.

automatically tuning a radio

As my loyal reader knows, Abbie Shaum (NYU) and I are building a software analog of an FM radio to find binary companions to stars with coherent oscillation modes. We have a good signal! But now we want to optimize the carrier frequency and some other properties of the radio. We discussed how to do that today.